Infinite Proper Subset of an Infinite Set

Date: 09/22/97 at 02:43:20
From: Sam Cotten
Subject: Infinite proper subset of an infinite set
My wife and I are in a Math survey class for Elementary school
teachers, (she teaches K, I teach 2nd grade.) A question on a recent
test was: Given set A= {1,2,3,...} and set B= {10,20,30,...}, is B a
proper subset of A?
The book we are using, _Fundamentals of Mathmatics_, by William M.
Setek, states that "A proper subset contains at least one less element
than the parent set." My reasoning was that since the two sets are
both "countable" infinite sets, and can be placed in a one-to-one
correspondence, they are therefore equal and B does not have one less
element; therefore B is not a proper subset of A.
I was marked wrong on this question, and the professor simply stated
that "B is contained in A, and is therefore a proper subset of A." I
guess my question has to do with the actual definition here: is the
book wrong, or at least worded improperly, or am I wrong in thinking
that the fact that two sets are the same size, i.e., they have the
same number of elements, precludes one from being a proper subset of
the other? It seems to me that it is merely a subset, not a proper
subset (?). Should the book not say something like "an element
not contained in..." rather than "...one less element"?

Date: 09/22/97 at 08:18:45
From: Doctor Jerry
Subject: Re: Infinite proper subset of an infinite set
Hi Sam,
Two sets can have the same number of elements and not be equal. The
set P={a,b,c,...,z} of 26 letters has the number of elements as the
set Q={1,2,3,...,26}, but the two sets have no elements in common.
The sets A= {1,2,3,...} and B= {10,20,30,...} have the same number of
elements (there is a one-to-one correspondence) and B is a (proper)
subset of A. The usual definition of subset is:
A set A is a subset of a set B if each element of A
is an element of B.
A set A is a proper subset of a set B if A is a subset of B and is not
equal to B.
I think the phrase "A proper subset contains at least one less element
than the parent set" is not widely used.
-Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Date: 09/26/97 at 12:57:17
From: Doctor Ken
Subject: Re: Infinite proper subset of an infinite set
Hi -
Let me add a little bit to this. Strictly speaking, if your textbook
only talks about finite sets, its definition is fine. But if it talks
about infinite sets (as your professor does) then it is incorrect, for
exactly the reason you stated: to properly apply the criterion to two
sets, you have to count up the number of elements in each set and
compare the two numbers you get. This technique fails in the case you
talked about.
A more suitable definition of proper subset would be something like
this:
A is a subset of B <=> every element of A is an element of B
A is a proper subset of B <=> A is a subset of B AND there is
some element of B that is not an element of A
or equivalently (as Dr. Jerry wrote)
A is a proper subset of B <=> A is a subset of B AND A does not
equal B
Under this more correct criterion, we can see fairly easily that your
set {10, 20, ...} is a proper subset of {1, 2, ...}. So it's quite
common for a set to be a proper subset of another set of the same
size.
Your professor should recognize that your book's definition is not
very good for infinite sets. At the very least it's ambiguous about
what's supposed to happen with infinite sets.
-Doctor Ken, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Date: 09/26/97 at 16:30:54
From: SAM COTTEN
Subject: Re: Infinite proper subset of an infinite set
Thanks very much for writing again. I talked to my professor about
this and was able to convey, though not as well as you have, what I
thought the problem in the book was. He understood my confusion, but
isn't going to give me credit for the question I missed as a result...
Interestingly, the book does not specifically address infinite sets in
this context, nor did he (the professor). It was just on the test,
and we were supposed to figure it out with the knowledge gained. I
think quite a few students shared my confusion. To him, and other
math-oriented people I have discussed this with, the answer is
obvious; so obvious that they didn't think they needed to explain it.
Perhaps the author of the book felt the same way.
Some of the interesting fallout of this situation is that: a) I had
to find out about this on my own - something we're always exhorting
our students to do - and b) now I understand a lot more about this
fascinating subject than I did before. So it's been well worth it
from that viewpoint.
I have not been in a pure math class for over 25 years, and a lot has
changed in the way math is taught. "New Math," as it was called then,
was introduced about halfway through my high school years, and I'm
afraid many of the teachers of that time were unable to really
comprehend it themselves, much less teach it effectively. I studied
set theory only briefly in one course (Algebra II?) in high school;
now, we use Venn diagrams in first grade onwards for everything from
math to language arts (they're great for comparing stories, etc.) I
only regret not getting this kind of foundation early on myself, as it
is becoming very clear how these concepts form the basis for all the
math I did take - and I took a lot of it! I'm only now beginning to
understand things I only did by rote then...
So thanks again, and keep up the good work, I'm very impressed with
your program!
Sam Cotten